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The Incidence Chromatic Number of Some Graph THE INCIDENCE CHROMATIC NUMBER OF SOME GRAPH LIU XIKUI AND LI YAN Received 1 April 2003 and in revised form 5 December 2003 The concept of the incidence chromatic number of a graph was introduced by Brualdi and Massey (1993). They conjectured that every graph G can be incidence colored with ∆(G) + 2 colors. In this paper, we calculate the incidence chromatic numbers of the com- plete k-partite graphs and give the incidence chromatic number of three infinite families of graphs. 1. Introduction Throughout the paper, all graphs dealt with are finite, simple, undirected, and loopless. Let G be a graph, and let V(G), E(G), ∆(G), respectively, denote vertex set, edge set, and maximum degree of G. In 1993, Brualdi and Massey [3] introduced the concept of incidence coloring. The order of G is the cardinality |v(G)|. The size of G is the cardinality |E(G)|.Let I(G) = (v,e) | v ∈ V, e ∈ E, v is incident with e (1.1) be the set of incidences of G. We say that two incidences (v,e)and(w, f )areadjacent provided one of the following holds: (i) v = w; (ii) e = f ; (iii) the edge vw = e or vw = f . Figure 1.1 shows three cases of two incidences being adjacent. An incidence coloring σ of G is a mapping from I(G)toasetC such that no two adjacent incidences of G have the same image. If σ : I(G) → C is an incidence coloring of G and |C|=k, k is a positive integer, then we say that G is k-incidence colorable. Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:5 (2005) 803–813 DOI: 10.1155/IJMMS.2005.803 804 The incidence chromatic number of some graph v v v ∗∗ ∗ ∗ efe e f w ∗ w ∗ v = w e = f vw = e (a) (b) (c) Figure 1.1. Cases of two incidences being adjacent. The minimum cardinality of C for which there exists an incidence coloring σ : I(G) → C is called the incidence chromatic number of G, and is denoted by inc(G). A partition {I1,I2,...,Ik} of I(G) is called an independence partition of I(G)ifeachIi is independent in I(G) (i.e., no two incidences of Ii are adjacent in I(G)). Clearly, for k ≥ inc(G), G is k-incidence colorable. We may consider G as a digraph by splitting each edge uv into two opposite arcs (u,v) and (v,u). Let e = uv.Weidentify(u,e)withthearc(u,v). So I(G) may be identified with the set of all arcs A(G). Two distinct arcs (incidences) (u,v)and(x, y)areadjacentifone of the following holds (see Figure 1.2): (1’) u = x; (2’) u = y and v = x; (3’) v = x. This concept was first developed by Brualdi and Massey [3] in 1993. They posed the incidence coloring conjecture (ICC), which states that for every graph G,inc(G) ≤ ∆ + 2. In 1997, Guiduli [5] showed that incidence coloring is a special case of directed star arboricity, introduced by Algor and Alon [1]. They pointed out that the ICC was solved in the negative following an example in [1]. Following the analysis in [1], they showed that inc(G) ≥ ∆ + Ω(log∆), where Ω = 1/8 −◦(1). Making use of a tight up- per bound for directed star arboricity, they obtained the upper bound inc(G) ≤ ∆ + O(log∆). Brualdi and Massey determined the incidence chromatic numbers of trees, complete graphs, and complete bipartite graphs [3]; Chen, Liu, and Wang determined the incidence chromatic numbers of paths, cycles, fans, wheels, adding-edge wheels, and complete 3- partite graphs [4]. In this paper, we will consider the incidence chromatic number for complete k-partite graphs. We will give the incidence chromatic number of complete k-partite graphs, and also give the incidence chromatic number of three infinite families of graphs. Let k be positive integer, put [k] = 1,2,...,k. We state first the following definitions. Definition 1.1. For a graph G(V,E)withvertexsetV and edge set E, the incidence graph I(G)ofG is defined as the graph with vertex set V(I(G)) and edge set E(I(G)). Definitions not given here may be found in [2]. L. Xikui and L. Yan 805 u(x) v(y) v(x) ∗∗ ∗ ∗ e ∗ ∗ vyu(x) u y (a) (b) (c) Figure 1.2. Cases of two arcs (incidences) are adjacent. 2. Some useful lemmas and properties of incidence chromatic number Lemma 2.1. Let T beatreeofordern ≥ 2 with maximum degree ∆. Then inc(T) = ∆ +1. Lemma 2.2. AgraphG is k-incidence colorable if and only if its incidence graph I(G) is k-vertex colorable, that is, inc(G) = χ(I(G)). Let M ={(ue,ve) | e = uv ∈ E(G), (ue,ve) ∈ E(I(G))},thenM forms a perfect match- ing of incidence graph I(G). The following lemmas are obvious. Lemma 2.3. The incidence graph I(G) of a graph G is a graph with a perfect matching. Lemma 2.4. For a graph G, v ∈ V(G),letBv ={(u,uv) | uv ∈ E(G), u ∈ V(G)}, Av = {(v,vu)|uv ∈ E(G), u ∈ V(G)}, then {Bv} is an independence-partition of incidence graph I(G), and the induced subgraph G[Av] of I(G) is a clique graph. By the definition of incidence graph, it is easy to give the proof. Lemma 2.5. Let ∆ be the maximum degree of graph G, I(G) the incidence graph, then com- plete graph K∆+1 is a subgraph of I(G). Proof. Let d(u) = ∆, p = ∆,andek = uv1, e2 = uv2, ..., ep = uv1 be the edges of G. p incidences in Iu ={(u,e1),(u,e2),...,(u,ep)} are adjacent to each other. For an incidence in Iv ={(vi,viu) | uvi ∈ G,1≤ i ≤ p},(vi,viu) is adjacent to all incidences in Iu.Sincep + 1incidences(u,e2),...,(u,ep),(vi,viu)areverticesofI(G), by the definition of incidence graph, we can complete the proof. Lemma 2.6. For a simple graph G with order n, inc(G) = n = ∆(G)+1, when ∆(G) = n − 1. Proof. Let |V(G)|=∆ +1= v(G), by Lemma 2.4, {Bv} is an independence-partition of incidence graph I(G), then χ(I(G)) ≤ v(G). By Lemma 2.5, K∆+1 is the subgraph of I(G), thus χ(I(G)) ≥ v(G), then inc(G) = χ(I(G)) = ∆ +1,asrequired. The following corollaries can be easily verified. Corollary 2.7. Let G be a graph with order n (n ≥ 2), then inc(G) ≤ ∆ +2, when ∆(G) = n − 2. 806 The incidence chromatic number of some graph In fact, for graphs G with order n, we can give each incidence in I(G)properinci- dence coloring as follows. Let V(G) ={v1,v2,...,vn} be the vertex set and C ={1,2,...,n} the color set. For i, j = 1,2,...,n,weletσ(vi,vivj ) = j. It is easy to see that the coloring above is an incidence coloring of G only with n colors. That is, inc(G) ≤ ∆ +2,when ∆ ≥ n − 2. Corollary 2.8. Let Wn be the wheel graph with order n +1. Then inc(Wn) = n +1. Lemma 2.9. Let H beasubgraphofG, then inc(H) ≤ G. Lemma 2.10. Let G be union of disjoint graphs G1,G2,..., and Gt.IfGi has an m-incidence coloring for all i = 1,2,...,t, then G has an m-incidence coloring. That is inc(G) = max{inc(Gi) | i = 1,2,...,t}. Proof. To prove this lemma, we only need to prove that G1 ∪ G2 has an m-incidence {I I ... I } I G {I I ... I } coloring. Let 1, 2, , m be an independence partition of ( 1), and 1, 2, , m I G {I ∪ I I ∪ I ... I ∪ I } an independence partition of ( 2). Then 1 1, 2 2, , m m forms an independence-partition of I(G1) ∪ I(G2). Hence G has an m-incidence coloring. The proof of the lemma is complete. Theorem 2.11. Let G be a graph with maximum degree ∆(G)=n−2 and minimum degree δ(G)≤[n/2]−1, then inc(G)=n−1=∆(G)+1. Proof. Let V(G) ={v1,v2,...,vn}, d(v1) = δ(G), and u/∈ V(G). Consider the auxiliary graph G with vertex set V(G ) = V(G) ∪{u} and edge set E(G ) = E(G) ∪{uvi | i = 2,3,...,n}. It follows that ∆(G ) = n − 1. Let G = G −{v1},then∆(G ) = n − 1, by Lemma 2.5,inc(G) = n.ForcolorsetC ={1,2,...,n}, suppose that σ is the n-incidence coloring of G with color set C. Without loss of generality, let σ (vi,vivj ) = j (vivj ∈ E(G)) and σ (vi,viu) = 1(i = 2,3,...,n), σ (u,uvi) = i (i = 2,3,...,n). In incidence set I(G), incidences (vi,vivj)(i, j = 2,3,...,n,andi = j)arealladjacentto(vi,viu)and (vj ,vj u), thus the color n cannot be used to color any incidence in I(G −{u}). De- N v ={v v ... v } v σ note by ( 1) i1 , i2 , , iδ the vertices adjacent to 1. The incidence coloring of graph G may be extended to an incidence coloring σ of graph G.Forx, y ∈ V(G)and x y/∈{v }∪N v σ x xy = σ x xy ∆ G = n − v k = , 1 ( 1), let ( , ) ( , ). Because ( ) 2, for vertex ik ( ... δ v ∈ V G v v ∈/ E G σ v v v = t 1,2, , ), there exists a vertex tk ( )suchthat ik tk ( ).
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